What are the domains where SVMs are still state-of-the-art?
State-of-the-art is a tough bar, because it's not clear how it should be measured. An alternative criteria, which is akin to state-of-the-art, is to ask when you might prefer to try an SVM.

SVMs have several advantages:


Through the kernel trick, the runtime of an SVM does not increase significantly if you want to learn patterns over many non-linear combinations of features, rather than the original feature set. In contrast, a more modern approach like a deep neural network will need to get deeper or wider to model the same patterns, which will increase its training time.
SVMs have an inherent bias towards picking "conservative" hypotheses, that are less likely to overfit the data, because they try to find maximum margin hypotheses. In some sense, they "bake-in" Occam's razor.
SVMs have only two hyperparameters (the choice of kernel and the regularization constant), so they are very easy to tune to specific problems. It is usually sufficient to tune them by performing a simple grid-search through the parameter space, which can be done automatically.


SVMs also have some disadvantages:


SVMs have a runtime that scales cubically in the number of datapoints you want to train on (i.e. $O(n^3)$ runtime)1. This does not compare well with, say, a typical training approach for a deep neural network which runs in $O(w*n*e)$ time, where $n$ is the number of data points, $e$ is the number of training epochs, and $w$ is the number of weights in the network. Generally $w, e &lt;&lt; n$.
To make use of the Kernel trick, SVMs cache a value for the kernelized "distance" between any two pairs of points. This means they need $O(n^2)$ memory. This is far, far, more trouble than the cubic runtime on most real-world sets. More than a few thousand datapoints will leave most modern servers thrashing, which increases effective runtime by several orders of magnitude. Together with point 1, this means SVMs will tend to become unworkably slow for sets beyond maybe 5,000-10,000 datapoints, at the upper limit. 


All of these factors point to SVMs being relevant for exactly one use case: small datasets where the target pattern is thought, apriori, to be some regular, but highly non-linear, function of a large number of features. This use case actually arises fairly often. A recent example application where I found SVMs to be a natural approach was building predictive models for a target function that was known to be the result of interactions between pairs of features (specifically, communications between pairs of agents). An SVM with a quadratic kernel could therefore efficiently learn conservative, reasonable, guesses.



1 There are approximate algorithms that will solve the SVM faster than this, as noted in the other answers.